Scale-Free Allocation, Amortized Convexity, and Myopic Weighted Paging

Inspired by Belady's optimal algorithm for unweighted paging, we consider a natural myopic model for weighted paging in which an algorithm has access to the relative ordering of all pages with respect to the time of their next arrival. We provide an $\ell$-competitive deterministic and an $O(\log \ell)$-competitive randomized algorithm, where $\ell$ is the number of distinct weight classes. Both these bounds are tight and imply an $O(\log W)$ and $O(\log \log W)$-competitive ratio respectively when the page weights lie between $1$ and $W$. Our model and results also simultaneously generalize the paging with predictions (Lykouris and Vassilvitskii, 2018) and the interleaved paging (Barve et al., 2000; Cao et al., 1994; Kumar et al., 2019) settings to the weighted case. Perhaps more interestingly, our techniques shed light on a family of interconnected geometric allocation problems. We show that the competitiveness of myopic weighted paging lies in between that of two similar problems called soft and strict allocation. While variants of the allocation problem have been studied previously in the context of the $k$-server problem, our formulation exhibits a certain scale-free property that poses significant technical challenges. Consequently, simple multiplicative weight update algorithms do not seem to work here; instead, we are forced to use a novel technique that decouples the rate of change of the allocation variables from their value. For offline unweighted paging, we also provide an alternate proof of the optimality of Belady's classic Farthest in Future algorithm, which may be of independent interest.